Edexcel C2 (Core Mathematics 2) 2005 January

Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 3 + 2 x ) ^ { 5 }\), giving each term in its simplest form.
(4)

The points \(A\) and \(B\) have coordinates \(( 5 , - 1 )\) and \(( 13,11 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\). Given that \(A B\) is a diameter of the circle \(C\),
2. find an equation for \(C\).

3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which

1. \(3 ^ { x } = 5\),
2. \(\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2\).

4. (a) Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ giving your answers to 1 decimal place where appropriate.

1. \(\quad \mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

When \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 1 .
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is 28 .

1. Find the value of \(a\) and the value of \(b\).
2. Show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).

1. The second and fourth terms of a geometric series are 7.2 and 5.832 respectively.

The common ratio of the series is positive.
For this series, find

1. the common ratio,
2. the first term,
3. the sum of the first 50 terms, giving your answer to 3 decimal places,
4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.

7. \begin{figure}[h] \end{figure} Figure 1 shows the triangle \(A B C\), with \(A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}\) and \(\angle B A C = 0.7\) radians. The \(\operatorname { arc } B D\), where \(D\) lies on \(A C\), is an arc of a circle with centre \(A\) and radius 8 cm . The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(B C\) and \(C D\) and the \(\operatorname { arc } B D\). Find

1. the length of the \(\operatorname { arc } B D\),
2. the perimeter of \(R\), giving your answer to 3 significant figures,
3. the area of \(R\), giving your answer to 3 significant figures.

8. \begin{figure}[h] \end{figure} The line with equation \(y = 3 x + 20\) cuts the curve with equation \(y = x ^ { 2 } + 6 x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.
2. Use calculus to find the exact area of \(S\).

9. \begin{figure}[h] \end{figure} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is \(2 x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2 x\) metres. The perimeter of the stage is 80 m .

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 } .$$
2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value.
3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\).
4. Calculate, to the nearest \(\mathrm { m } ^ { 2 }\), the maximum area of the stage.